Báo cáo hóa học: " Research Article Subordination and Superordination for Multivalent Functions Associated with the Dziok-Srivastava Operator" doc

Volume 2011, Article ID 486595, 17 pagesdoi:10.1155/2011/486595 Research Article Subordination and Superordination for Multivalent Functions Associated with the Dziok-Srivastava Operator

Volume 2011, Article ID 486595, 17 pages

doi:10.1155/2011/486595

Research Article

Subordination and Superordination for

Multivalent Functions Associated with

the Dziok-Srivastava Operator

1 Department of Applied Mathematics, Pukyong National University, Busan 608-737, Republic of Korea

2 Department of Mathematics, Kyungsung University, Busan 608-736, Republic of Korea

3 School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Penang, Malaysia

4 Department of Mathematics, University of Delhi, Delhi 110007, India

Correspondence should be addressed to Rosihan M Ali,rosihan@cs.usm.my

Received 21 September 2010; Revised 18 January 2011; Accepted 26 January 2011

Academic Editor: P J Y Wong

Copyrightq 2011 Nak Eun Cho et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

Subordination and superordination preserving properties for multivalent functions in the open unit disk associated with the Dziok-Srivastava operator are derived Sandwich-type theorems for these multivalent functions are also obtained

1 Introduction

Let : {z ∈  :|z| < 1} be the open unit disk in the complex plane, and letH : H  denote the class of analytic functions defined in For n ∈  : {1, 2, } and a ∈ , let

Ha, n consist of functions f ∈ H of the form fz  a  an z n  an1z n1 · · · Let f and F

be members ofH The function f is said to be subordinate to F, or F is said to be superordinate

to f, if there exists a function w analytic in , with |wz| ≤ |z| and such that fz  Fwz.

In such a case, we write f ≺ F or fz ≺ Fz If the function F is univalent in , then f ≺ F if and only if f0  F0 and f  ⊂ F  cf 1,2 Let ϕ :

2 → , and let h be univalent in The subordination ϕpz, zpz ≺ hz is called a first-order differential subordination.

It is of interest to determine conditions under which p ≺ q arises for a prescribed univalent function q The theory of differential subordination in is a generalization of a differential inequality in, and this theory of differential subordination was initiated by the works of Miller, Mocanu, and Reade in 1981 Recently, Miller and Mocanu3 investigated the dual

problem of differential superordination The monograph by Miller and Mocanu 1 gives a good introduction to the theory of differential subordination, while the book by Bulboac˘a 4 investigates both subordination and superordination Related results on superordination can

be found in5 23

By using the theory of differential subordination, various subordination preserving properties for certain integral operators were obtained by Bulboac˘a24, Miller et al 25, and Owa and Srivastava26 The corresponding superordination properties and sandwich-type results were also investigated, for example, in4 In the present paper, we investigate subordination and superordination preserving properties of functions defined through the

use of the Dziok-Srivastava linear operator Hp,q,sα1 see 1.9 and 1.10, and also obtain corresponding sandwich-type theorems

The Dziok-Srivastava linear operator is a particular instance of a linear operator

defined by convolution For p∈, letApdenote the class of functions

f z  z p∞

k1

a k p z k p 1.1

that are analytic and p-valent in the open unit disk with f p1 0 / 0 The Hadamard

productor convolution f ∗ g of two analytic functions

f z ∞

k0

a k z k , g z ∞

k0

b k z k 1.2

is defined by the series



f ∗ gz ∞

k0

a k b k z k 1.3

For complex parameters α1, , α q and β1, , β s βj /  0, −1, −2, ; j  1, , s, the

generalized hypergeometric function q F sα1, , α q; β1, , β s; z is given by

q F s



α1, , α q; β1, , β s; z

:∞

n0

α1n· · ·α q

n



β1



n· · ·β s



n

z n

n! ,



q ≤ s  1; q, s ∈ 0 :∪ {0}; z ∈ ,

1.4

where ν n is the Pochhammer symbol or the shifted factorial defined in terms of the Gamma function by

ν n: Γν  n

Γν 

⎧

⎨

⎩

1 if n  0, ν ∈ \ {0},

ν ν  1 · · · ν  n − 1 if n ∈ , ν∈ . 1.5

To define the Dziok-Srivastava operator

H p

α1, , α q ; β1, , β s

:Ap → Ap 1.6 via the Hadamard product given by1.3, we consider a corresponding function

Fpα1, , α q ; β1, , β s ; z

1.7 defined by

Fpα1, , α q ; β1, , β s ; z

: zp

q F s

α1, , α q ; β1, , β s ; z

. 1.8 The Dziok-Srivastava linear operator is now defined by the Hadamard product

H p

α1, , α q ; β1, , β s

f z : Fpα1, , α q ; β1, , β s ; z

∗ fz. 1.9

This operator was introduced and studied in a series of recent papers by Dziok and Srivastava

27–29; see also 30,31 For convenience, we write

H p,q,sα1 : Hpα1, , α q ; β1, , β s



The importance of the Dziok-Srivastava operator from the general convolution operator rests

on the relation

z

H p,q,sα1fz α1H p,q,sα1 1fz −α1− pH p,q,sα1fz 1.11

that can be verified by direct calculations see, e.g., 27 The linear operator Hp,q,sα1 includes various other linear operators as special cases These include the operators introduced and studied by Carlson and Shaffer 32, Hohlov 33, also see 34,35, and Ruscheweyh36, as well as works in 27,37

2 Definitions and Lemmas

Recall that a domain D ⊂  is convex if the line segment joining any two points in D lies entirely in D, while the domain is starlike with respect to a point w0∈ D if the line segment joining any point in D to w0 lies inside D An analytic function f is convex or starlike if

f is, respectively, convex or starlike with respect to 0 For f ∈ A : A1, analytically, these functions are described by the conditions Re1  zfz/fz > 0 or Rezfz/fz >

0, respectively More generally, for 0 ≤ α < 1, the classes of convex functions of order α and starlike functions of order α are, respectively, defined by Re1  zfz/fz > α or

Rezfz/fz > α A function f is close-to-convex if there is a convex function g not

necessarily normalized such that Refz/gz > 0 Close-to-convex functions are known

to be univalent

The following definitions and lemmas will also be required in our present investiga-tion

Definition 2.1see 1, page 16 Let ϕ :

2 → , and let h be univalent in If p is analytic in

and satisfies the differential subordination

ϕ

p z, zpz≺ hz, 2.1

then p is called a solution of differential subordination 2.1 A univalent function q is called

a dominant of the solutions of differential subordination 2.1, or more simply a dominant, if

p ≺ q for all p satisfying 2.1 A dominant q that satisfies q ≺ q for all dominants q of 2.1 is said to be the best dominant of2.1

Definition 2.2see 3, Definition 1, pages 816-817 Let ϕ : 

2 → , and let h be analytic in If p and ϕpz, zpz are univalent in and satisfy the differential superordination

h z ≺ ϕp z, zpz, 2.2

then p is called a solution of differential superordination 2.2 An analytic function q is

called a subordinant of the solutions of differential superordination 2.2, or more simply

a subordinant, if q ≺ p for all p satisfying 2.2 A univalent subordinant q that satisfies q ≺ q for all subordinants q of2.2 is said to be the best subordinant of 2.2

Definition 2.3see 1, Definition 2.2b, page 21 Denote by Q the class of functions f that are analytic and injective on \ Ef, where

E

f



ζ ∈ ∂ : lim

z → ζ f z  ∞ , 2.3

and are such that fζ / 0 for ζ ∈ ∂ \ Ef.

Lemma 2.4 cf 1, Theorem 2.3i, page 35 Suppose that the function H :

2 →  satisfies the condition

Re His, t ≤ 0, 2.4

for all real s and t ≤ −n1  s2/2, where n is a positive integer If the function pz  1  pn z n · · ·

is analytic in and

Re H

p z, zpz> 0 z ∈ , 2.5

then Re p z > 0 in

One of the points of importance ofLemma 2.4was its use in showing that every convex function is starlike of order 1/2see e.g., 38, Theorem 2.6a, page 57 In this paper, we take

an opportunity to use the technique in the proof ofTheorem 3.1

Lemma 2.5 see 39, Theorem 1, page 300 Let β, γ ∈  with β /  0, and let h ∈ H  with

h 0  c If Reβhz  γ > 0 for z ∈ , then the solution of the differential equation

q z  zqz

βq z  γ  hz z ∈  2.6

with q 0  c is analytic in and satisfies Reβqz  γ > 0 z ∈ .

Lemma 2.6 see 1, Lemma 2.2d, page 24 Let p ∈ Q with p0  a, and let qz  aanz n· · ·

be analytic in with q z /≡ a and n ≥ 1 If q is not subordinate to p, then there exists points z0 

r0e iθ ∈ and ζ0∈ ∂ \ Ep, for which q r0 ⊂ p ,

q z0  pζ0, z0qz0  mζ0pζ0 m ≥ n. 2.7

A function Lz, t defined on × 0, ∞ is a subordination chain or L¨owner chain if

L ·, t is analytic and univalent in for all t ∈ 0, ∞, Lz, · is continuously differentiable on

0, ∞ for all z ∈ , and Lz, s ≺ Lz, t for 0 ≤ s < t.

Lemma 2.7 see 3, Theorem 7, page 822 Let q ∈ Ha, 1, ϕ : 

2 → , and set h z ≡

ϕ qz, zqz If Lz, t  ϕqz, tzqz is a subordination chain and p ∈ Ha, 1 ∩ Q, then

h z ≺ ϕp z, zpz 2.8

implies that

q z ≺ pz. 2.9

Furthermore, if ϕ qz, zpz  hz has a univalent solution q ∈ Q, then q is the best subordinant.

Lemma 2.8 see 3, Lemma B, page 822 The function Lz, t  a1tz  · · · , with a1t / 0 and

limt→ ∞|a1t|  ∞, is a subordination chain if and only if

Re

z∂L z, t/∂z

∂L z, t/∂t

> 0 z ∈ ; 0 ≤ t < ∞. 2.10

3 Main Results

We first prove the following subordination theorem involving the operator Hp,q,sα1 defined

by1.10

Theorem 3.1 Let f, g ∈ A p For α1> 0, 0 ≤ λ < p, let

ϕ z : p − λ

p

H p,q,sα1 1gz

p

H p,q,sα1gz

z p z ∈ . 3.1

Suppose that

Re

1 zϕz

ϕz

> −δ, z ∈ , 3.2

where

δ



p − λ2 p2α2

1− p − λ2− p2α2

1

4p

p − λα1 . 3.3

Then the subordination condition

p − λ

p

H p,q,sα1 1fz

p

H p,q,sα1fz

z p ≺ ϕz 3.4

implies that

H p,q,sα1fz

z p ≺H p,q,sα1gz

Moreover, the function H p,q,sα1gz/z p is the best dominant.

Proof Let us define the functions F and G, respectively, by

F z : H p,q,sα1fz

z p , G z : H p,q,sα1gz

z p 3.6

We first show that if the function q is defined by

q z : 1  zGz

Gz , 3.7

then

Re qz > 0 z ∈ . 3.8

Logarithmic differentiation of both sides of the second equation in 3.6 and using

1.11 for g ∈ Apyield

pα1

p − λ ϕ z 

pα1

p − λ G z  zGz. 3.9

Now, differentiating both sides of 3.9 results in the following relationship:

1zϕz

ϕz  1 

zGz

Gz 

zqz

q z  pα1/

p − λ

 qz  zqz

q z  pα1/

p − λ  ≡ hz.

3.10

We also note from3.2 that

Re

h z  pα1

p − λ

and, by usingLemma 2.5, we conclude that differential equation 3.10 has a solution q ∈ H  with q0  h0  1 Let us put

H u, v  u  v

u  pα1/

p − λ   δ, 3.12 where δ is given by3.3 From 3.2, 3.10, and 3.12, it follows that

Re

H

q z, zqz> 0 z ∈ . 3.13

In order to useLemma 2.4, we now proceed to show that Re His, t ≤ 0 for all real s and t ≤ −1  s2/2 Indeed, from 3.12,

Re His, t  Re



is  pα1/

p − λ   δ



 tpα1/

p − λ

pα1/ p − λ  is2  δ

≤ − E δs

2 pα1/

p − λ is 2,

3.14

where

E δs :

pα

1

p − λ − 2δ

s2− pα1

p − λ

2δ pα1

p − λ− 1

. 3.15

For δ given by3.3, we can prove easily that the expression Eδs given by 3.15 is positive or equal to zero Hence, from3.14, we see that Re His, t ≤ 0 for all real s and

t ≤ −1  s2/2 Thus, by usingLemma 2.4, we conclude that Re qz > 0 for all z ∈ That is,

G defined by3.6 is convex in Next, we prove that subordination condition 3.4 implies that

F z ≺ Gz 3.16

for the functions F and G defined by3.6 Without loss of generality, we also can assume that

G is analytic and univalent on and Gζ / 0 for |ζ|  1 For this purpose, we consider the function Lz, t given by

L z, t : Gz 



p − λ1  t

pα1 zGz z ∈ ; 0 ≤ t < ∞. 3.17

Note that

∂L z, t

∂z z0 G0



pα1p − λ1  t

pα1



/

 0 0≤ t < ∞; α1> 0; 0 ≤ λ < p. 3.18

This shows that the function

L z, t  a1tz  · · · 3.19

satisfies the condition a1t / 0 for all t ∈ 0, ∞ Furthermore,

Re

z∂L z, t/∂z

∂L z, t/∂t

 Re

pα

1

p − λ  1  t

1zGz

Gz

> 0. 3.20

Therefore, by virtue ofLemma 2.8, Lz, t is a subordination chain We observe from

the definition of a subordination chain that

L ζ, t / ∈ L , 0  ϕ  ζ ∈ ∂ ; 0 ≤ t < ∞. 3.21

Now suppose that F is not subordinate to G; then, byLemma 2.6, there exist points z0 ∈

and ζ0∈ ∂ such that

F z0  Gζ0, z0F z0  1  tζ0Gζ0 0 ≤ t < ∞. 3.22

L ζ0, t   Gζ0 



p − λ1  t

pα1 ζ0G

ζ0

 Fz0 



p − λ

pα1 z0F

z0

 p − λ

p

H p,q,sα1 1fz0

p

H p,q,sα1fz0

z p0 ∈ ϕ ,

3.23

by virtue of subordination condition 3.4 This contradicts the above observation that

L ζ0, t  / ∈ ϕ  Therefore, subordination condition 3.4 must imply the subordination given

by3.16 Considering Fz  Gz, we see that the function G is the best dominant This

evidently completes the proof ofTheorem 3.1

We next prove a dual result to Theorem 3.1, in the sense that subordinations are replaced by superordinations

Theorem 3.2 Let f, g ∈ A p For α1> 0, 0 ≤ λ < p, let

ϕ z : p − λ

p

H p,q,sα1 1gz

p

H p,q,sα1gz

z p z ∈ . 3.24

Suppose that

Re

1 zϕz

ϕz

> −δ, z ∈ , 3.25

where δ is given by3.3 Further, suppose that

p − λ

p

H p,q,sα1 1fz

p

H p,q,sα1fz

is univalent in and H p,q,sα1fz/z p ∈ H1, 1 ∩ Q Then the superordination

ϕ z ≺ p − λ

p

H p,q,sα1 1fz

p

H p,q,sα1fz

z p 3.27

implies that

H p,q,sα1gz

z p ≺ H p,q,sα1fz

Moreover, the function H λ,q,sα1gz/z p is the best subordinant.

Proof The first part of the proof is similar to that ofTheorem 3.1and so we will use the same notation as in the proof ofTheorem 3.1

Now let us define the functions F and G, respectively, by3.6 We first note that if the

function q is defined by3.7, then 3.9 becomes

ϕ z  Gz  p − λ

pα1 zGz. 3.29 After a simple calculation,3.29 yields the relationship

1zϕz

ϕz  qz  zqz

q z  pα1/

p − λ  3.30

Then by using the same method as in the proof ofTheorem 3.1, we can prove that Re qz > 0 for all z ∈ That is, G defined by 3.6 is convex univalent in Next, we prove that the subordination condition3.27 implies that

G z ≺ Fz 3.31

for the functions F and G defined by3.6 Now considering the function Lz, t defined by

L z, t : Gz 



p − λt

pα1

zGz z ∈ ; 0 ≤ t < ∞, 3.32

we can prove easily that Lz, t is a subordination chain as in the proof of Theorem 3.1 Therefore according toLemma 2.7, we conclude that superordination condition3.27 must imply the superordination given by 3.31 Furthermore, since the differential equation

3.29 has the univalent solution G, it is the best subordinant of the given differential

superordination This completes the proof ofTheorem 3.2

Combining Theorems3.1and3.2, we obtain the following sandwich-type theorem

Theorem 3.3 Let f, g k∈ Apk  1, 2 For k  1, 2, α1> 0, 0 ≤ λ < p, let

ϕ kz : p − λ

p

H p,q,sα1 1gkz

p

H p,q,sα1gkz

z p z ∈ . 3.33

Suppose that

Re



1zϕk z

ϕk z



> −δ, 3.34

where δ is given by3.2 Further, suppose that

p − λ

p

H p,q,sα1 1fz

p

H p,q,sα1fz

Proof The first part of the proof is similar to that ofTheorem 3. 1and so we will use the same notation as in the proof ofTheorem 3.1

Now let us define the. ..

for the functions F and G defined by3.6 Without loss of generality, we also can assume that

G is analytic and univalent on and Gζ / for |ζ|  For this... a subordination chain as in the proof of Theorem 3.1 Therefore according toLemma 2.7, we conclude that superordination condition3.27 must imply the superordination given by 3.31 Furthermore,